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- Deformed Laguerre-Hahn orthogonal polynomials on the real linePublication . Branquinho, A.; Rebocho, M. N.We study families of orthogonal polynomials on the real line whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients. We derive discrete dynamical systems, obtained as a result of deformations of the recurrence relation coefficients of the orthogonal polynomials related to the above referred Stieltjes functions.
- Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circlePublication . Branquinho, A.; Rebocho, M. N.In this paperwe characterize sequences of orthogonal polynomials on the unit circle whose Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of matrix Sylvester differential equations. For the particular case of semi-classical orthogonal polynomials on the unit circle, it is derived a characterization in terms of first order linear differential systems.
- Sylvester equations for Laguerre-Hahn orthogonal polynomials on the real linePublication . Branquinho, A.; Paiva, Anabela; Rebocho, M. N.Matrix Sylvester differential equations are introduced in the study of Laguerre-Hahn orthogonal polynomials. Matrix Sylvester differential systems are shown to yield representations for the Laguerre-Hahn orthogonal polynomials. Lax pairs are given, formed from the differential system and the recurrence relation, that yield discrete non-linear equations for the three term recurrence relation coefficients of the Laguerre-Hahn orthogonal polynomials.